Group #28336630 — subgroups

Subgroups — #28336630

Order 13696.

256 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812042	C₂	1	2	yes	maximal
2	C₂×#16812040	C₂	1	2	yes	maximal
3	C₂×#16812020	C₂	1	1	yes	maximal, characteristic
4	#20292497	C₂	1	1	yes	maximal, characteristic
5	#382×C₁₀₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₀₇⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₁₀₇	—	2	2	no	—
8	C₂×C₄⋊Dic₁₀₇	C₂²	1	2	yes	—
9	#16812042	—	2	4	no	—
10	#16812040	—	2	4	no	—
11	#16812020	C₄	1	4	yes	complemented
12	C₂×C₄⋊C₄×C₁₀₇	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₁₀₇	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
15	#95×C₁₀₇	—	2	2	no	—
16	C₂²×Dic₁₀₇	—	2	2	no	—
17	C₂×C₁₀₇⋊C₈	C₂×C₄	1	4	yes	—
18	C₂×C₁₀₇⋊C₈	D₄	1	4	yes	—
19	C₂×C₁₀₇⋊C₈	Q₈	1	4	yes	—
20	C₄⋊Dic₁₀₇	—	2	2	no	—
21	C₄⋊Dic₁₀₇	—	2	2	no	—
22	C₄⋊Dic₁₀₇	—	4	4	no	—
23	C₂²×C₄×C₁₀₇	—	2	1	no	—
24	C₂²×C₄×C₁₀₇	—	2	1	no	—
25	C₂²×C₄×C₁₀₇	D₄	1	2	yes	—
26	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₁₀₇	—	2	2	no	—
28	C₄⋊C₄×C₁₀₇	—	2	2	no	—
29	C₄²×C₁₀₇	C₂×C₄	1	2	yes	—
30	C₄²×C₁₀₇	Q₈	1	2	yes	—
31	C₂³×C₁₀₇	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₁₀₇	—	2	2	no	—
33	C₂×C₄×C₁₀₇	—	2	2	no	—
34	C₂×C₄×C₁₀₇	—	4	1	no	—
35	C₂×C₄×C₁₀₇	—	4	1	no	—
36	C₂×C₄×C₁₀₇	—	4	2	no	—
37	C₂×C₄×C₁₀₇	—	4	2	no	—
38	C₂×C₄×C₁₀₇	C₄⋊C₄	1	4	yes	—
39	C₂×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
41	C₂×C₄×C₁₀₇	C₂×Q₈	1	2	yes	—
42	C₂×C₄×C₁₀₇	C₂×Q₈	1	1	yes	characteristic
43	C₂×Dic₁₀₇	—	4	2	no	—
44	C₂×Dic₁₀₇	—	4	4	no	—
45	C₁₀₇⋊C₈	C₄⋊C₄	1	8	yes	complemented
46	C₂²×C₁₀₇	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₁₀₇	#127	1	1	yes	characteristic
48	C₂²×C₁₀₇	C₄⋊Q₈	1	1	yes	characteristic
49	C₂²×C₁₀₇	AGL(1,ℤ/8)	1	2	yes	—
50	C₂²×C₁₀₇	#137	1	2	yes	—
51	C₄×C₁₀₇	—	2	4	no	—
52	C₄×C₁₀₇	—	8	1	no	—
53	C₄×C₁₀₇	—	8	1	no	—
54	C₄×C₁₀₇	C₂×C₄⋊C₄	1	2	yes	—
55	C₄×C₁₀₇	C₄⋊Q₈	1	2	yes	—
56	Dic₁₀₇	—	8	2	no	—
57	C₂×C₁₀₇	#384	1	1	yes	characteristic
58	C₂×C₁₀₇	#428	1	2	yes	—
59	C₂×C₁₀₇	#496	1	1	yes	characteristic
60	C₂×C₁₀₇	#497	1	1	yes	characteristic
61	C₂×C₁₀₇	#501	1	2	yes	—
62	#1847	—	107	1	no	maximal, Sylow
63	C₁₀₇	#1847	1	1	yes	characteristic, complemented, Sylow
64	C₂×#107	—	107	2	no	—
65	C₂×#106	—	107	2	no	—
66	C₂×#97	—	107	1	no	—
67	#384	—	107	1	no	—
68	#382	D₁₀₇	1	1	yes	characteristic
69	C₂²×C₈	—	107	2	no	—
70	C₂×C₄⋊C₄	—	107	2	no	—
71	C₂×C₄⋊C₄	—	214	2	no	—
72	C₂×C₄⋊C₄	C₂×D₁₀₇	1	1	yes	characteristic
73	C₂×C₄⋊C₄	C₂×D₁₀₇	1	1	yes	characteristic
74	C₂×C₄²	C₂×D₁₀₇	1	1	yes	characteristic
75	#107	—	214	4	no	—
76	#106	—	214	4	no	—
77	#97	—	107	4	no	—
78	#95	—	2	2	no	—
79	C₂²×C₄	—	2	1	no	—
80	C₂²×C₄	—	2	1	no	—
81	C₂²×C₄	—	214	2	no	—
82	C₂²×C₄	C₂²⋊D₁₀₇	1	2	yes	—
83	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
84	C₂×C₈	—	107	4	no	—
85	C₂×C₈	—	107	4	no	—
86	C₂×C₈	—	107	4	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	428	4	no	—
90	C₄⋊C₄	—	214	2	no	—
91	C₄⋊C₄	—	214	2	no	—
92	C₄²	Dic₂₁₄	1	2	yes	—
93	C₄²	C₄×D₁₀₇	1	2	yes	—
94	C₂³	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	4	1	no	—
98	C₂×C₄	—	4	1	no	—
99	C₂×C₄	—	4	2	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	428	2	no	—
102	C₂×C₄	—	428	4	no	—
103	C₂×C₄	Dic₁₀₇⋊C₄	1	4	yes	—
104	C₂×C₄	C₂×C₄×D₁₀₇	1	1	yes	characteristic
105	C₂×C₄	D₄×D₁₀₇	1	2	yes	—
106	C₂×C₄	Q₈×D₁₀₇	1	2	yes	—
107	C₂×C₄	C₂×Dic₂₁₄	1	1	yes	characteristic
108	C₈	—	107	8	no	—
109	C₂²	C₂×Dic₁₀₇⋊C₄	1	1	yes	characteristic
110	C₂²	#16812105	1	1	yes	characteristic
111	C₂²	#16812107	1	1	yes	characteristic
112	C₂²	#16812134	1	2	yes	—
113	C₂²	#16812139	1	2	yes	—
114	C₄	—	2	4	no	—
115	C₄	—	8	1	no	—
116	C₄	—	8	1	no	—
117	C₄	—	856	2	no	—
118	C₄	C₄⋊C₄×D₁₀₇	1	2	yes	—
119	C₄	#16812108	1	2	yes	—
120	C₂	#20292496	1	1	yes	characteristic
121	C₂	#20292670	1	2	yes	—
122	C₂	#20293017	1	1	yes	characteristic
123	C₂	#20293022	1	1	yes	characteristic
124	C₂	#20293037	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812042

C₂

yes

maximal

C₂×#16812040

C₂

yes

maximal

C₂×#16812020

C₂

yes

maximal, characteristic

#20292497

C₂

yes

maximal, characteristic

#382×C₁₀₇

C₂

yes

maximal, characteristic

C₂²×C₁₀₇⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₁₀₇

—

C₂×C₄⋊Dic₁₀₇

C₂²

yes

—

#16812042

—

#16812040

—

#16812020

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₀₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₀₇

C₂²

yes

characteristic

C₂×C₄²×C₁₀₇

C₂²

yes

characteristic

#95×C₁₀₇

—

C₂²×Dic₁₀₇

—

C₂×C₁₀₇⋊C₈

C₂×C₄

yes

—

C₂×C₁₀₇⋊C₈

D₄

yes

—

C₂×C₁₀₇⋊C₈

Q₈

yes

—

C₄⋊Dic₁₀₇

—

C₄⋊Dic₁₀₇

—

C₄⋊Dic₁₀₇

—

C₂²×C₄×C₁₀₇

—

C₂²×C₄×C₁₀₇

—

C₂²×C₄×C₁₀₇

D₄

yes

—

C₂²×C₄×C₁₀₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₇

—

C₄⋊C₄×C₁₀₇

—

C₄²×C₁₀₇

C₂×C₄

yes

—

C₄²×C₁₀₇

Q₈

yes

—

C₂³×C₁₀₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂×D₄

yes

—

C₂×C₄×C₁₀₇

C₂×Q₈

yes

—

C₂×C₄×C₁₀₇

C₂×Q₈

yes

characteristic

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

C₁₀₇⋊C₈

C₄⋊C₄

yes

complemented

C₂²×C₁₀₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₀₇

#127

yes

characteristic

C₂²×C₁₀₇

C₄⋊Q₈

yes

characteristic

C₂²×C₁₀₇

AGL(1,ℤ/8)

yes

—

C₂²×C₁₀₇

#137

yes

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

C₂×C₄⋊C₄

yes

—

C₄×C₁₀₇

C₄⋊Q₈

yes

—

Dic₁₀₇

—

C₂×C₁₀₇

#384

yes

characteristic

C₂×C₁₀₇

#428

yes

—

C₂×C₁₀₇

yes

C₂×C₁₀₇

yes

C₂×C₁₀₇

yes

—

#1847

—

107

maximal, Sylow

C₁₀₇

#1847

yes

characteristic, complemented, Sylow

C₂×#107

—

107

—

C₂×#106

—

107

—

C₂×#97

—

107

—

#384

—

107

—

#382

D₁₀₇

yes

characteristic

C₂²×C₈

—

107

—

C₂×C₄⋊C₄

—

107

—

C₂×C₄⋊C₄

—

214

—

C₂×C₄⋊C₄

C₂×D₁₀₇

yes

characteristic

C₂×C₄⋊C₄

C₂×D₁₀₇

yes

characteristic

C₂×C₄²

C₂×D₁₀₇

yes

characteristic

#107

—

214

—

#106

—

214

—

#97

—

107

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

214

—

C₂²×C₄

C₂²⋊D₁₀₇

yes

—

C₂²×C₄

C₂²×D₁₀₇

yes

characteristic

C₂×C₈

—

107

—

C₂×C₈

—

107

—

C₂×C₈

—

107

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

428

—

C₄⋊C₄

—

214

—

C₄⋊C₄

—

214

—

C₄²

Dic₂₁₄

yes

—

C₄²

C₄×D₁₀₇

yes

—

C₂³

C₂×C₂²⋊D₁₀₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

428

—

102

C₂×C₄

—

428

—

103

C₂×C₄

Dic₁₀₇⋊C₄

yes

—

104

C₂×C₄

C₂×C₄×D₁₀₇

yes

characteristic

105

C₂×C₄

D₄×D₁₀₇

yes

—

106

C₂×C₄

Q₈×D₁₀₇

yes

—

107

C₂×C₄

C₂×Dic₂₁₄

yes

characteristic

108

C₈

—

107

—

109

C₂²

C₂×Dic₁₀₇⋊C₄

yes

characteristic

110

C₂²

#16812105

yes

characteristic

111

C₂²

#16812107

yes

characteristic

112

C₂²

#16812134

yes

—

113

C₂²

#16812139

yes

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

856

—

118

C₄

C₄⋊C₄×D₁₀₇

yes

—

119

C₄

#16812108

yes

—

120

C₂

#20292496

yes

characteristic

121

C₂

#20292670

yes

—

122

C₂

#20293017

yes

characteristic

123

C₂

#20293022

yes

characteristic

124

C₂

#20293037

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₀₇
commutator	C₂×C₄×C₁₀₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₇
Fitting	#382×C₁₀₇	C₂
socle, prime radical	C₂³×C₁₀₇	C₂×D₄
cosocle	C₂²×C₄×C₁₀₇	C₂³
hypercenter	#382	D₁₀₇
hypocenter	C₁₀₇	#1847
LCS	G ▷ C₂×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₀₇

commutator

C₂×C₄×C₁₀₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₇

Fitting

#382×C₁₀₇

C₂

socle, prime radical

C₂³×C₁₀₇

C₂×D₄

cosocle

C₂²×C₄×C₁₀₇

C₂³

D₁₀₇

C₁₀₇

G ▷ C₂×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

N	H ≅ G/N	type
11. #16812020	C₄	semidirect
45. C₁₀₇⋊C₈	C₄⋊C₄	semidirect
63. C₁₀₇	#1847	semidirect

H ≅ G/N

type

11. #16812020

C₄

semidirect

45. C₁₀₇⋊C₈

C₄⋊C₄

semidirect

63. C₁₀₇

#1847

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#16812042	2
C₂×#16812040	2
C₂×#16812020	2
#20292497	2
#382×C₁₀₇	2
#1847	107

subgroup

index

#382×C₁₀₇

107

Subgroups — #28336630

Conjugacy classes of subgroups

Complements

Maximal subgroups